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73
RoundRobin Scheduling for MaxMin Fairness in Data Networks
 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS
, 1991
"... This paper studies a simple strategy, proposed independently by Gallager [1] and Katevenis [2], for fairly allocating link capacity in a pointtopoint packet network with virtual circuit routing. Each link offers its packet transmission slots to its user sessions by polling them in roundrobin orde ..."
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Cited by 109 (0 self)
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This paper studies a simple strategy, proposed independently by Gallager [1] and Katevenis [2], for fairly allocating link capacity in a pointtopoint packet network with virtual circuit routing. Each link offers its packet transmission slots to its user sessions by polling them in roundrobin order. In addition, window flow control is used to prevent excessive packet queues at the network nodes. As the window size increases, the session throughput rates are shown to approach limits that are perfectly fair in the maxmin sense. That is, the smallest session rate in the network is as large as possible and, subject to that constraint, the secondsmallest session rate is as large as possible, etc. If each session has periodic input (perhaps with jitter) or has such heavy demand that packets are always waiting to enter the network, then a finite window size suffices to produce perfectly fair throughput rates. The roundrobin method is considerably simpler than earlier strategies for achieving global fairness. The fair session rates are not explicitly computed, and the only overhead communication is that required for the window acknowledgments. The main drawback is that large windows are needed to achieve even approximately fair throughputs in some (hopefully rare) situations, and large windows permit large crossnetwork delays. Fortunately, the roundrobin method offers other throughput guarantees that, while falling short of perfect fairness, do apply even for sessions with small windows. Such sessions are promised reasonable bounds on their crossnetwork packet delay as well.
The Second Eigenvalue of the Google Matrix
, 2003
"... We determine analytically the modulus of the second eigenvalue for the web hyperlink matrix used by Google for computing PageRank. Specifically, we prove the following statement: "For any matrix A = [cP + (1 , where P is an n n rowstochastic matrix, E is a nonnegative nn rankone rowst ..."
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Cited by 70 (8 self)
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We determine analytically the modulus of the second eigenvalue for the web hyperlink matrix used by Google for computing PageRank. Specifically, we prove the following statement: "For any matrix A = [cP + (1 , where P is an n n rowstochastic matrix, E is a nonnegative nn rankone rowstochastic matrix, and 0 1, the second eigenvalue of A has modulus #2  # c. Furthermore, if P has at least two irreducible closed subsets, the second eigenvalue #2 = c." This statement has implications for the convergence rate of the standard PageRank algorithm as the web scales, for the stability of PageRank to perturbations to the link structure of the web, for the detection of Google spammers, and for the design of algorithms to speed up PageRank.
Rigorous Hitting Times for Binary Mutations
, 1999
"... In the binary evolutionary optimization framework, two mutation operators are theoretically investigated. For both the standard mutation, in which all bits are flipped independently with the same probability, and the 1bitflip mutation, which flips exactly one bit per bitstring, the statistical dis ..."
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Cited by 60 (2 self)
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In the binary evolutionary optimization framework, two mutation operators are theoretically investigated. For both the standard mutation, in which all bits are flipped independently with the same probability, and the 1bitflip mutation, which flips exactly one bit per bitstring, the statistical distribution of the first hitting times of the target are thoroughly computed (expectation and variance) up to terms of order l (the size of the bitstrings) in two distinct situations: without any selection, or with the deterministic (1+1)ES selection on the OneMax problem. In both cases, the 1bitflip mutation convergence time is smaller by a constant (in terms of l) multiplicative factor. These results extend to the case of multiple independent optimizers. Keywords Evolutionary algorithms, stochastic analysis, binary mutations, Markov chains, hitting times. 1 Introduction One known drawback of Evolutionary Algorithms as function optimizers is the amount of computational efforts they re...
Strong Uniform Times and Finite Random Walks
 ADVANCES IN APPLIED MATHEMATICS 8,6997 (1987)
, 1987
"... There are several techniques for obtaining bounds on the rate of convergence to the stationary distribution for Markov chains with strong symmetry properties, in particular random walks on finite groups. An elementary method, strong uniform times, is often effective. We prove such times always exist ..."
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Cited by 59 (7 self)
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There are several techniques for obtaining bounds on the rate of convergence to the stationary distribution for Markov chains with strong symmetry properties, in particular random walks on finite groups. An elementary method, strong uniform times, is often effective. We prove such times always exist, and relate this method to coupling and Fourier analysis.
Finite Markov Chain Results in Evolutionary Computation: A Tour d'Horizon
, 1998
"... . The theory of evolutionary computation has been enhanced rapidly during the last decade. This survey is the attempt to summarize the results regarding the limit and finite time behavior of evolutionary algorithms with finite search spaces and discrete time scale. Results on evolutionary algorithms ..."
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Cited by 57 (2 self)
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. The theory of evolutionary computation has been enhanced rapidly during the last decade. This survey is the attempt to summarize the results regarding the limit and finite time behavior of evolutionary algorithms with finite search spaces and discrete time scale. Results on evolutionary algorithms beyond finite space and discrete time are also presented but with reduced elaboration. Keywords: evolutionary algorithms, limit behavior, finite time behavior 1. Introduction The field of evolutionary computation is mainly engaged in the development of optimization algorithms which design is inspired by principles of natural evolution. In most cases, the optimization task is of the following type: Find an element x 2 X such that f(x ) f(x) for all x 2 X , where f : X ! IR is the objective function to be maximized and X the search set. In the terminology of evolutionary computation, an individual is represented by an element of the Cartesian product X \Theta A, where A is a possibly...
How Mutation and Selection Solve Long Path Problems in Polynomial Expected Time
, 1996
"... It is shown by means of Markov chain analysis that unimodal binary long path problems can be solved by mutation and elitist selection in a polynomially bounded number of trials on average. 1 Unimodality of Binary Functions The notion of unimodal functions usually appears in the theory of optimizati ..."
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Cited by 52 (2 self)
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It is shown by means of Markov chain analysis that unimodal binary long path problems can be solved by mutation and elitist selection in a polynomially bounded number of trials on average. 1 Unimodality of Binary Functions The notion of unimodal functions usually appears in the theory of optimization in IR 1 . Elster et al. (1977), pp. 228230, provide a precise definition that is specialized to functions in IR 1 whereas the definition in Bronstein and Semendjajew (1988), p. 137, for functions in IR ` with ` 1 presupposes differentiability. Here, the following definition for functions over IB ` will be used: Definition 1 Let f be a realvalued function with domain IB ` where IB = f0; 1g. A point x 2 IB ` is called a local solution of f if f(x ) f(x) for all x 2 fy 2 IB ` : k y \Gamma x k 1 = 1g (1) where k x k 1 = P ` i=1 j x i j is the Hamming norm. If the inequality in (1) is strict, then x is termed a strictly local solution. The value f(x ) at a...
Towards an analytic framework for analysing the computation time of evolutionary algorithms
 Artificial Intelligence
, 2003
"... In spite of many applications of evolutionary algorithms in optimisation, theoretical results on the computation time and time complexity of evolutionary algorithms on different optimisation problems are relatively few. It is still unclear when an evolutionary algorithm is expected to solve an optim ..."
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Cited by 36 (13 self)
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In spite of many applications of evolutionary algorithms in optimisation, theoretical results on the computation time and time complexity of evolutionary algorithms on different optimisation problems are relatively few. It is still unclear when an evolutionary algorithm is expected to solve an optimisation problem efficiently or otherwise. This paper gives a general analytic framework for analysing first hitting times of evolutionary algorithms. The framework is built on the absorbing Markov chain model of evolutionary algorithms. The first step towards a systematic comparative study among different EAs and their first hitting times has been made in the paper.
Convergence Properties of Some MultiObjective Evolutionary Algorithms
 IN CONGRESS ON EVOLUTIONARY COMPUTATION (CEC 2000
, 2000
"... We present four abstract evolutionary algorithms for multiobjective optimization and theoretical results that characterize their convergence behavior. Thanks to these results it is easy to verify whether a particular instantiation of these abstract evolutionary algorithms offers the desired limit b ..."
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Cited by 35 (5 self)
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We present four abstract evolutionary algorithms for multiobjective optimization and theoretical results that characterize their convergence behavior. Thanks to these results it is easy to verify whether a particular instantiation of these abstract evolutionary algorithms offers the desired limit behavior or not. Several examples are given.
Time complexity of evolutionary algorithms for combinatorial optimization: A decade of results
 International Journal of Automation and Computing
, 2007
"... Abstract: Computational time complexity analyses of Evolutionary Algorithms (EAs) have been performed since the midnineties. The first results were related to very simple algorithms, such as the (1+1)EA, on toy problems. These efforts produced a deeper understanding of how EAs perform on different ..."
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Cited by 22 (10 self)
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Abstract: Computational time complexity analyses of Evolutionary Algorithms (EAs) have been performed since the midnineties. The first results were related to very simple algorithms, such as the (1+1)EA, on toy problems. These efforts produced a deeper understanding of how EAs perform on different kinds of fitness landscapes and general mathematical tools that may be extended to the analysis of more complicated EAs on more realistic problems. In fact, in recent years, it has been possible to analyse the (1+1)EA on combinatorial optimization problems with practical applications and more realistic populationbased EAs on structured toy problems. This paper presents a survey of the results obtained in the last decade along these two research lines. The most common mathematical techniques are introduced, the basic ideas behind them are discussed and their elective applications are highlighted. Solved problems that were still open are enumerated as are those still awaiting for a solution. New questions and problems arisen in the meantime are also considered. Keywords: Evolutionary algorithms, computational complexity, combinatorial optimization, evolutionary computation theory.